A002376 Least number of positive cubes needed to sum to n.

1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 1, 2, 3, 4, 5, 4, 5, 6, 2, 3, 4, 5, 6, 5, 6, 7, 3, 4, 5, 6, 7, 6, 7, 8, 4, 5, 6, 2, 3, 4, 5, 6, 5, 6, 7, 3, 4, 1, 2, 3, 4, 5, 6, 4, 5, 2, 3, 4, 5, 6, 7, 5, 6, 3, 3, 4, 5, 6, 7, 6, 7, 4, 4, 5, 2, 3, 4, 5, 6, 5, 5, 6, 3, 4, 5, 6, 7, 6, 6

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

No terms are greater than 9, see A002804. - Charles R Greathouse IV, Aug 01 2013

References

• D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 81.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• A. R. Zornow, De compositione numerorum e cubis integris positivus, J. Reine Angew. Math., 14 (1835), 276-280.

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Eric Weisstein's World of Mathematics, Cubic Number

Crossrefs

Cf. A000578, A003325 (numbers requiring 2 cubes), A047702 (numbers requiring 3 cubes), A047703 (numbers requiring 4 cubes), A047704 (numbers requiring 5 cubes), A046040 (numbers requiring 6 cubes), A018890 (numbers requiring 7 cubes), A018888 (numbers requiring 8 or 9 cubes), A055401 (cubes needed by greedy algorithm).

Programs

Maple

f:= proc(n) option remember;
  min(seq(procname(n - i^3)+1, i=1..floor(n^(1/3))))
end proc:
f(0):= 0:
map(f, [$1..100]); # Robert Israel, Jun 30 2017

Mathematica

CubesCnt[n_] := Module[{k = 1}, While[Length[PowersRepresentations[n, k, 3]] == 0, k++]; k]; Array[CubesCnt, 100] (* T. D. Noe, Apr 01 2011 *)

Python

from itertools import count
from sympy.solvers.diophantine.diophantine import power_representation
def A002376(n):
    if n == 1: return 1
    for k in count(1):
        try:
            next(power_representation(n,3,k))
        except:
            continue
        return k # Chai Wah Wu, Jun 25 2024

Formula

The g.f. conjectured by Simon Plouffe in his 1992 dissertation,

-(-1-z-z^2-z^3-z^4-z^5-z^6+6*z^7)/(z+1)/(z^2+1)/(z^4+1)/(z-1)^2, is incorrect: the first wrong coefficient is that of z^26. - Robert Israel, Jun 30 2017

Extensions

More terms from Arlin Anderson (starship1(AT)gmail.com)